These fluctuations are
constrained by fundamental symmetry relations, known as fluctuation
relations (FRs) (see
Refs. [1]}, [2]}, [3]}, [4]}, [5]} for
collections and reviews). Applied to the (total)
entropy production, they allow for negative entropy production on
the trajectory level but reduce to the conventional second law of
thermodynamics (expressing positiveness of the entropy production)
upon averaging. FRs were put forward by Evans et
al. [6]} in numerical simulations of
shear-driven systems, but later mathematically proven for different
dynamics [7]}, [8]}, [9]} and also experimentally confirmed
[10]}, [11]}, [12]}, [13]}, [14]}, [2]}, [4]}.
More generally, FRs relate the probability density functions of
certain thermodynamic observables to those of conjugate
(typically time-reversed) processes. An important example
in the FR collection is the Jarzynski
relation [17]}, [18]}
involving the non-equilibrium work of driven systems, which is of
great importance because of its applicability for measuring free
energy landscapes [13]}. Subsequently, many other
associated relations have been discovered, such as the Crooks
fluctuation relation [20]}, [21]}, the
Hummer-Szabo relation [22]}, and integral FRs
(IFRs) [10]}, [24]}, [25]}.
